Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

EXISTENCE AND UNIQUENESS OF ENTROPY SOLUTION TO PRESSURELESS EULER SYSTEM WITH A FLOCKING DISSIPATION

We study the existence and uniqueness problem for the nonhomogeneous pressureless Euler system with the initial density being a Radon measure. Our uniqueness result is obtained in the same space as the existence theorem. Besides, by counterexample we prove that Huang-Wang’s energy condition is also necessary for our nonhomogeneous system.

Uniqueness of BV Entropy Solutions for Higher Dimensional Quasilinear Parabolic Equations with Arbitrary Degeneracy

We are concerned with the uniqueness of solutions of the Cauchy problemand a(s),b(s) are appropriately smooth.Since a(s) is allowed to have zero points, we call them points of degeneracy of (1), the equation (1) does not admit classical solutions in general. The solutions of (1) even might be discontinuous, whenever the set E = {s : a(s) = 0} includes interior points.Equations with degeneracy arise from a wide variety of diffusive processes in nature

EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ（T ） is assumed to have a positive lower bound. We first consider the Cauchy problem （the initial value problem）, that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an ad-ditional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave （weak or strong, determined by the wedge angle at the wedge vertex） when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.

On Existence of Entropy Solution for a Doubly Nonlinear Differential Equation with L1 -Data

We consider a class of doubly nonlinear history-dependent problems having a convection term and a pseudomonotone nonlinear diffusion operator associated an equation of the type ?t(k * (b(v) - b(v0))) - div(a(x,Dv) + F(v)) = f where the right hand side belongs to L1. The kernel k belongs to the large class of PC kernels. In particular, the case of fractional time derivatives of order α ∈ (0,1) is included. Assuming b nondecreasing with L1-data, we prove existence in the framework of entropy solutions. The approach adopted for the proof is based on a several step approximation method and by using a result in the case of a strictly increasing b.

Global Entropy Solutions of the Cauchy Problem for Nonhomogeneous Relativistic Euler System

We analyze the 2 × 2 nonhomogeneous relativistic Euler equations for perfect fluids in special relativity. We impose appropriate conditions on the lower order source terms and establish the existence of global entropy solutions of the Cauchy problem under these conditions.

Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems Via a Sequence of Penalized Equations

We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.

Global Weak Entropy Solution of Nonlinear Ideal Reaction Chromatography System and Applications

The ideal reaction chromatography model can be regarded as a semi-coupled system of two hyperbolic partial differential equations, in which, one is a self-closed nonlinear equation for the reactant concentration and another is a linear equation coupling the reactant concentration for the resultant concentration. This paper is concerned with the initial-boundary value problem for the above model. By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial initial-boundary value problem for Riemann type of initial-boundary data. Moreover, as examples, we apply the obtained results to the cases of head-on and wide pulse injections and give the expression of the global weak entropy solution.

The Existence of Entropy Solutions to Some Parabolic Problems with L^1 Data

In this paper, an existence result of entropy solutions to some parabolic problems is established. The data belongs to L^1 and no growth assumption is made on the lower-order term in divergence form.

Stability of Entropy Solution of Conservation Law Systems on Flow Function and Relaxation Function

In this paper,using the fractional step Lax-Friedrichs difference scheme,we establish the stability of the entropy solution on flow function and relaxation function for a class of conservation law systems with a source term and a relaxation term.

L1 Existence and Uniqueness of Entropy Solutions to Nonlinear Multivalued Elliptic Equations with Homogeneous Neumann Boundary Condition and Variable Exponent

In this work, we study the following nonlinear homogeneous Neumann boundary value problemβ （u） -diva （x, 7u） f in fΩ, a （x, u）. η= 0 on Ω, where Ω is a smooth bounded open domain in RN, N ≥ 3 with smooth boundary Ωand ηthe outer unit normal vector on Ω . We prove the existence and uniqueness of an entropy solution for L1-data f. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.

Large Time Behavior of Entropy Solutions to Some Hyperbolic System with Dissipative Structure

In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t → ∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.

One-Sided Exact Boundary Null Controllability of Entropy Solutions to a Class of Hyperbolic Systems of Conservation Laws with Characteristics with Constant Multiplicity

This paper proves the local exact one-sided boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws with characteristics with constant multiplicity. This generalizes the results in [Li, T. and Yu, L., One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws, To appear in Journal de Mathematiques Pures et Appliquees, 2016.] for a class of strictly hyperbolic systems of conservation laws.

Weak Solutions of a Class of Strongly Degenerate Quasilinear Parabolic Equations

In this paper, we show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.

New Method of Predicting the Thermodynamic Properties of Solutions

Thermodynamic functions solutions of a 25 binary systems formed n-alcohols and esters of aliphatic acids by were calculated using the standards ideal solution and ideal gas. The value change regularity of the thermodynamic functions of solutions depending on their molar mass and concentration of the solutions＇ components was determined by the thermodynamic analysis. The method of prediction of the thermodynamic properties of binary solutions was suggested on the basis of the determined regularities. The corresponding equations were obtained.

Entropy and Renormalized Solutions for Nonlinear Elliptic Problem Involving Variable Exponent and Measure Data

We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.

THE CARBUNCLE PHENOMENON IS INCURABLE

Numerical approximations of multi-dimensional shock waves sometimes ex- hibit an instability called the carbuncle phenomenon. Techniques for suppressing carbuncles are trial-and-error and lack in reliability and generality, partly because theoretical knowledge about carbuncles is equally unsatisfactory. It is not known which numerical schemes are affected in which circumstances, what causes carbuncles to appear and whether carbuncles are purely mimerical artifacts or rather features of a continuum equation or model. This article presents evidence towards the latter： we propose that carbuncles are a special class of entropy solutions which can be physically correct in some circumstances. Using ＂filaments＂, we trigger a single carbuncle in a new and more reliable way, and compute the structure in detail in similarity coordinates. We argue that carbuncles can, in some circumstances, be valid vanishing viscosity limits. Trying to suppress them is making a physical assumption that may be false.

THE HOMOGENEOUS DIRICHLET PROBLEM FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC-HYPERBOLIC EQUATION WITH L^p INITIAL VALUE

The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p initial value.We use the device of doubling variables and some technical analysis to prove the uniqueness result.Moreover we can prove that the L p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.

THE APPLICATION OF THE KINETIC FORMULATION TO A SYSTEM OF QUADRATIC FLUX

In this article, the author uses the compensated compactness method coupled with some basic ideas of the kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global entropy solutions to a system of quadratic flux. The fire-new method of reduction of Young measures is a pith of this work.

THE FORMATION OF SHOCK WAVES OF THE EQUATIONS OF MAGNETOHYDRODYNAMICS

The property of fluid field of one-dimensional magnetohydrodynamics (MHD) transverse flow after the appearance of singularity is discussed. By the method of iteration, the strong discontinuity (shock wave) and entropy solution are constructed and the estimations on the singularity of the solution near the point of blow-up are obtained.